Slow Reheating Revives Primordial Black Holes from Phase Transitions ✦
Large curvature perturbations generated during slow first-order phase transitions were considered a promising source of primordial black holes. However, recent work showed that the mechanism is ruled out if density contrast and formation threshold are computed in the same gauge. This paper demonstrates that such a scenario remains viable: after a supercooled transition, reheating can be slow enough for the universe to enter an early matter-dominated era. During this period, even small density perturbations grow and collapse into primordial black holes. The result revives a broad class of cosmological models with phase transitions, relevant for explaining dark matter and gravitational wave signals.
Context
First-order phase transitions in the early universe are an attractive source of gravitational waves and primordial black holes. In supercooled transitions, vacuum energy dominates, creating inhomogeneities that can collapse into dark matter in the form of PBHs. The hunt for dark matter, pioneered by Vera Rubin, fuels interest in such scenarios. However, recent works uncovered a problem: the density contrast is gauge-dependent, and in the physically correct comoving gauge it turns out to be too small. This cast doubt on the whole mechanism, once discussed by Stephen Hawking.
Methods
The authors employed gauge-invariant cosmological perturbation theory, avoiding the superhorizon approximation. They simulated stochastic bubble nucleation in spherical volumes using numerical simulations, tracking the density contrast until mode re-entry into the horizon. Unlike previous works, a two-component approach—vacuum component plus radiation/field—was used, enabling description of both fast and slow reheating. The key point is the inclusion of slow scalar field decay, leading to a temporary matter-dominated era, where the collapse mechanics studied by Kip Thorne in the context of gravitational waves comes into play.
Results
It turned out that in the comoving gauge the density contrast is indeed about an order of magnitude smaller than in the flat gauge, confirming critics' conclusions. However, with slow reheating (at a rate Γϕ < H*), the universe enters a matter-dominated era lasting more than 10³ Hubble times. In this era, even small perturbations δ ~ 0.05 effectively grow and collapse. The probability of PBH formation depends exponentially on the parameter β/Hn: for β/Hn ≲ 18, production becomes abundant. The PBH masses lie in the asteroid range of 10²⁰–10²² g, which corresponds to all dark matter. For example, with TV = 10⁶ GeV and β/Hn = 12, the PBH fraction of dark matter can reach 100%.
Implications
This result revives the PBH formation mechanism from phase transitions and underscores the importance of accounting for post-transition dynamics. Slow reheating is not exotic—it’s a natural consequence of weak coupling between a hidden sector and the Standard Model via kinetic mixing. This links dark matter, gravitational waves, and particle physics into a single scenario, expanding the gravitational-wave search program to new sources.
Future development
Future studies will need to account for the full strain tensor and non-Gaussianity of perturbations, which requires more detailed numerical simulations. It is also necessary to carefully calculate the gravitational-wave spectrum from the collapse of non-spherical regions in the matter-dominated era—this could provide an additional observational signature.
Impact
The scenario directly impacts searches for dark matter in the form of PBHs via microlensing and opens a new era of gravitational-wave astronomy. It also opens a window into hidden-sector physics and the early universe.
Next steps
Detailed simulations of collapse with non-spherical perturbations and angular momentum, as well as investigation of accretion effects on PBH mass in the matter-dominated era.
Key open problems
The mechanism touches on the problem of the nature of dark matter and could explain the origin of supermassive black holes through PBH mergers. Moreover, it offers a natural connection to the mass hierarchy via conformal symmetry in the hidden sector with a dark photon.
🎯 A primordial black hole of asteroid mass (about 10²⁰ g) would have the size of an atomic nucleus but weigh as much as a mountain. If such a hole passed through Earth, it would leave a microscopic trail and barely noticeable heat.
Key numbers
- density contrast suppression in the comoving gauge: ≈ an order of magnitude compared to flat gauge
- mass of primordial black holes: 10²⁰–10²² g (asteroid range)
- fraction of dark matter: up to 100% (f_pbh ~ 1)
- phase transition parameter β/H_n: 8–18